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## Homework Statement

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I am trying to get the C-G Decomposition for 6 ⊗ 3.

2. Homework Equations

2. Homework Equations

Neglecting coefficients a tensor can be decomposed into a symmetric part and an antisymmetric part. For the 6 ⊗ 3 = (2,0) ⊗ (1,0) this is:

T

^{ij}⊗ T

^{k}= Q

^{ijk}= (Q

^{{ij}k}+ Q

^{{ji}k}) + (Q

^{[ij]k}+ Q

^{[ji]k})

Where the focus is only on the interchange of the i and j indeces.

3. The Attempt at a Solution

3. The Attempt at a Solution

Consider the antisymmetric term: (Q

^{[ij]k}+ Q

^{[ji]k}). Use the invariant tensor to get:

ε

^{ijl}{ε

_{lmn}Q

^{mnk}+ ε

_{lnm}Q

^{nmk}) = ε

^{ijl}(Q

^{k}

_{l}+ Q

^{k}

_{l})

So the symmetric part is the '10' (3,0) and the antisymmetric part is the '8' (1,1). The symmetric part is traceless. However, I think I have neglected the trace of the antisymmetric term Q

^{k}

_{l}and should be writing Q

^{k}

_{l}- δ

^{k}

_{l}Q. However, If I do this I now have to add the singlet which shouldn't be there. What am I going wrong?